Probabilities from entanglement, Born's rule p~k=b¿~kb^2 from envariance (29 pages)

Wojciech Hubert Zurek

Ph. Blanchard

Wojciech Hubert Zurek

I show how probabilities arise in quantum physics by exploring the implications of environment-assisted invariance or envariance, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like'' states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Born's rule ${p}_{k}\ensuremath{\propto}{\ensuremath{\mid}{\ensuremath{\psi}}_{k}\ensuremath{\mid}}^{2}$ for the outcomes associated with Schmidt states. The probabilities derived in this manner are an objective reflection of the underlying state of the system---they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge'' of the observer. This envariance-based approach is compared with and found to be superior to prequantum definitions of probability including the standard definition based on the ``principle of indifference'' due to Laplace and the relative frequency approach advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Born's rule: Envariance is enough to establish the dynamical independence of preferred branches of the evolving state vector of the composite system and, thus, to arrive at the environment-induced superselection (einselection) of pointer states, which was usually derived by an appeal to decoherence. The envariant origin of Born's rule for probabilities sheds light on the relation between ignorance (and, hence, information) and the nature of quantum states.

F.J. Duarte

Fedor Herbut

Zurek's derivation of Born's rule using envariance (invariance due to entanglement) is considered to capture the probability in full generality, but only as applied to measurement of a quantum observable. Contrariwise, textbook formulations of Born's rule begin with a pure state of a closed, undivided system. The task of this study is to show that a rearrangement of the Zurek approach is possible in which the latter is viewed as giving the probabilities for Schmidt states of an arbitrary composite state vector, and afterwards it is extended to probabilities in a closed, undivided system. This is achieved by determining simultaneously probability and measurement based on the fact that the physical meaning of probability and that of measurement are inextricably dependent on each other.

Alexander Rudol'fovich Its, V. E. Korepin